Make the dough: In a small saucepan, heat milk and butter over medium-low heat until butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C).
In the bowl of a stand mixer, whisk together 3 cups (375 grams) flour, sugar, yeast, salt, orange zest, and anise seed. Add warm milk mixture, room temperature eggs, and almond extract; using the paddle attachment, beat at low speed until combined. With mixer on medium-low speed, gradually add 1¾ cups (219 grams) flour, beating until a soft dough forms.
Switch to the dough hook attachment. Beat at medium-low speed until dough is smooth and elastic, about 15 minutes; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Turn out dough onto a lightly floured surface, and shape into a round.
Lightly spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover with plastic wrap. Refrigerate for at least 8 hours, preferably overnight. (Alternatively, cover and let rise in a warm, draft-free place [75°F/24°C] until doubled in size, about 1 hour. Punch down dough, and refrigerate for 30 minutes. It will be easier to braid while cold.)
Line a baking sheet with parchment paper.
Prepare eggs: Lightly coat in-shell eggs with vegetable oil. Set aside.
Shape the Tsoureki: Punch dough down, and lightly knead 2 to 3 times. Divide into 3 portions (about 389 grams each). Roll each portion into a 26-inch rope. Pinch strands together on one end, and braid strands together, pinching to seal on other end. Shape into a wreath, tucking ends under wreath. Transfer to prepared pan. Gently tuck oiled in-shell eggs between strands of dough in wreath as desired. Cover with plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until puffed, 45 minutes to 1 hour.
Preheat oven to 350°F (180°C).
Topping: In a small bowl, lightly whisk together 1 tablespoon (15 grams) water and room temperature egg. Lightly brush egg wash over braid, avoiding in-shell eggs. Sprinkle sliced almonds evenly over braid.
Bake: Bake until golden and an instant-read thermometer inserted in center registers 190°F (88°C), about 50 minutes, covering with foil during final 15 minutes of baking to prevent excess browning. Let cool on pan for 15 minutes. Remove from pan and let cool completely on a wire rack.
Carefully remove in-shell eggs from loaf. Clean off any bread or vegetable oil residue with a clean towel.
Dye eggs: In a medium bowl, whisk together warm water, vinegar, and food coloring until combined. Carefully lower in-shell eggs into bowl; let stand until desired color is reached, about 45 minutes to 1 hour, stirring occasionally. Remove in-shell eggs from water, and transfer to a paper towel to dry for at least 30 minutes. Carefully place in-shell eggs back in loaf. Serve immediately. Store bread (with eggs removed, store these in refrigerator) in an airtight container at room temperature for up to 3 days.